Let Y be a shifted exponential random variable with density...

Let $Y$ be a shifted exponential random variable with density function given by $$ f(s)=\alpha e^{-\alpha(s-D)}, \quad \text { for } s \geq D . $$ Prove the following: $$ \begin{aligned} Y^*[\theta] & =e^{-\theta D} \times \frac{\alpha}{\alpha+\theta}, \\ E[Y] & =D+\frac{1}{\alpha}, \\ \operatorname{Var}[Y] & =\frac{1}{\alpha^2} \end{aligned} $$ and $$ F(x)=P[Y \leq x]=1-e^{-\alpha(x-D)}, \text { for } x \geq D . $$

Let $Y$ be a shifted exponential random variable with density function given by
$$
f(s)=\alpha e^{-\alpha(s-D)}, \quad \text { for } s \geq D .
$$
Prove the following:
$$
\begin{aligned}
Y^*[\theta] & =e^{-\theta D} \times \frac{\alpha}{\alpha+\theta}, \\
E[Y] & =D+\frac{1}{\alpha}, \\
\operatorname{Var}[Y] & =\frac{1}{\alpha^2}
\end{aligned}
$$
and
$$
F(x)=P[Y \leq x]=1-e^{-\alpha(x-D)}, \text { for } x \geq D .
$$

Key Concepts

Shifted Exponential Distribution

A shifted exponential distribution is a variant of the standard exponential distribution where the support starts at a value D rather than 0. This shift parameter D translates the entire distribution, so probabilities and moments are computed accordingly, reflecting the fact that the random variable takes values only from D onward.

Laplace Transform and Moment Generating Functions

The Laplace transform or moment generating function is a fundamental tool for summarizing the distribution of a random variable by integrating its density function multiplied by an exponential function. In the context of the shifted exponential, one uses a substitution to account for the shift D, yielding expressions such as e^(-?D) multiplied by ?/(?+?). This approach efficiently captures the moment information of the distribution.

Expectation and Variance Calculation

Expectation and variance are key moments that describe the central tendency and spread of the random variable, respectively. For the shifted exponential, the expected value is calculated by adding the shift D to the mean of the standard exponential component (which is 1/?), and the variance remains that of the underlying exponential distribution, namely 1/?^2, since the shift does not affect the dispersion.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable gives the probability that the variable is less than or equal to a given value. For a shifted exponential distribution, the CDF is obtained by integrating the density function starting from the shift parameter D, resulting in an expression of the form 1 - e^(-?(x-D)) for values x ? D. This reflects how the probability accumulates starting from the shifted origin.

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